# Intra-class correlation

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In statistics, the intraclass correlation (or the intraclass correlation coefficient[1]) is a measure of correlation, consistency or conformity for a data set when it has multiple groups. There are several measures of ICC and they may yield different values for the same data set.[2]

## Early definition Edit

Consider a data set with two groups represented in a data matrix $X(N' \times 2)$ then the intraclass correlation r is computed from[3]

$\bar{x} = \frac{1}{2N'} \sum_{n=1}^{N'} (x_{n,1} + x_{n,2})$,
$s^2 = \frac{1}{2N} \left\{ \sum_{n=1}^{N} ( x_{n,1} - \bar{x})^2 + \sum_{n=1}^{N} ( x_{n,2} - \bar{x})^2 \right\}$,
$r = \frac{1}{Ns^2} \sum_{n=1}^{N} ( x_{n,1} - \bar{x}) ( x_{n,2} - \bar{x})$,

where N is the degree of freedoms (Note that the precise form of the formula differ between versions of Fisher's book: The 1954 edition[3] uses $N'$ in places where the 1925 edition[4] uses $N$). This form is not the same as the interclass correlation. For the data set with two groups the intraclass correlation r will be confined to the interval [-1, +1].

The intraclass correlation is also defined for data sets with more than two groups, e.g., for three groups it is computed as[3]

$\bar{x} = \frac{1}{3 N'} \sum_{n=1}^{N'} (x_{n,1} + x_{n,2} + x_{n,2})$,
$s^2 = \frac{1}{3 N} \left\{ \sum_{n=1}^{N} ( x_{n,1} - \bar{x})^2 + \sum_{n=1}^{N} ( x_{n,2} - \bar{x})^2 + \sum_{n=1}^{N} ( x_{n,3} - \bar{x})^2\right\}$,
$r = \frac{1}{3Ns^2} \sum_{n=1}^{N} \left\{ ( x_{n,1} - \bar{x})( x_{n,2} - \bar{x}) + (x_{n,1} - \bar{x})( x_{n,3} - \bar{x})+( x_{n,2} - \bar{x})( x_{n,3} - \bar{x}) \right\}$.

(Also this form differs between editions of Fisher's book)

As the number of groups grow, the number of terms in the form will grow exponentially, but another form has been suggested that does not require so many computations[3]

$K\sum_{k=1}^{K} ( \bar{x}_k - \bar{x})^2 = Ns^2 \left\{1+(K-1) r \right\}$,

where K is the number of groups. This form is usually attributed to Harris.[5] The left term is non-negative, consequently the intraclass correlation must be

$r \geq -1 /(K-1)$.

## "Modern" ICCs Edit

Beginning with Ronald Fisher the intraclass correlation has been regarded within the framework of analysis of variance (ANOVA). Different ICCs arise with different ANOVA models, e.g., one-way analysis or two-way analysis, and they may produce marked different results. An article by McGraw and Wong lists these variations.[6]

Yet another measure that has been regarded as an intraclass correlation coefficient is the concordance correlation coefficient.

## References Edit

There is an entire chapter that concerns the intraclass correlation in Ronald Fisher's classic book Statistical Methods for Research Workers[3].

### Notes Edit

1. Koch, Gary G. "Intraclass correlation coefficient". Encyclopedia of Statistical Sciences 4. Ed. Samuel Kotz and Norman L. Johnson. New York: John Wiley & Sons. 213–217.

2. Reinhold Müller & Petra Büttner (December 1994). A critical discussion of intraclass correlation coefficients. Statistics in Medicine 13 (23-24): 2465–2476.
3. 3.0 3.1 3.2 3.3 3.4 Ronald A. Fisher (1954). Statistical Methods for Research Workers, Twelfth edition, Oliver and Boyd.
4. Ronald A. Fisher (1925). Statistical Methods for Research Workers, Oliver and Boyd.
5. J. Arthur Harris (October 1913). On the Calculation of Intra-Class and Inter-Class Coefficinets of Correlation from Class Moments when the Number of Possible Combinations is Large 9 (3/4): 446–472.
6. Kenneth O. McGraw & S. P. Wong (1996). Forming inferences about some intraclass correlation coefficients. Psychological Methods 1: 30–46. There are several errors in the article: